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Section 4

The Quadratic Formula and Applications

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$x^{2}+4 x-11=0$$

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$3 x^{2}+7 x-8=0$$

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$5 x^{2}-8 x-12=0$$

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$7 x+3 x^{2}-10=0$$

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$-27 x+10-5 x^{2}=0$$

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$4 x=9 x^{2}+4$$

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$3-2 x=8 x+5 x^{2}-11$$

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$2 x^{2}=10$$

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$5 x^{2}=-11$$

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$3 x=7 x^{2}$$

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$4 x^{2}=9 x$$

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$(x-2)(3 x+4)=5$$

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$(2 x-5)(4 x+3)=7$$

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$4(3 x-4)(5 x-6)=25$$

Rewrite the equation in the form $a x^{2}+b x+c=0,$ with $a > 0,$ and then determine the coefficients $a, b,$ and $c .$ Do not solve the equation.$$(x-7)(2 x-3)=3 x(2 x-4)$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}+2 x-10=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}+6 x+3=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}-2 x-10=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}-6 x+3=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}+6 x=3$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}-8 x=4$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}-12 x-4=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}+5 x=5$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}-5 x=5$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}-3 x-1=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}+2 x+10=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$(x-4)(x+1)=6$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$(x-3)(x+5)=20$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$(2 x-3)(3 x+1)=-4$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$(x-3)(x+5)=12$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$(x+3)(x+2)=-10$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$(2 x-3)(3 x+5)=-20$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}-6 x+23=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}-12 x=-48$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}+10 x+50=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}-8 x+36=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}-4 x=8$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}+6 x+21=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}-3 x+3=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$x^{2}+5 x+10=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$2 x^{2}+8 x+1=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$2 x^{2}+x-6=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$6 x^{2}+5 x-6=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$12 x^{2}-7 x=12$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$6 x^{2}+23 x=-20$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$2 x^{2}+5 x+10=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$4 x^{2}-5 x+8=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$1 / 2 x^{2}+5 x+2=0$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$\frac{1}{2} x^{2}-\frac{3}{4} x=\frac{5}{6}$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$\frac{3}{5} x^{2}-\frac{3}{2} x=\frac{7}{10}$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$3-4 x^{2}=8 x$$

In each of the following, solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth.$$9-3 x^{2}+5 x=8+2 x^{2}$$

The sum of the squares of three consecutive integers is $110,$ find the integers.

The sum of the squares of three consecutive odd integers is $515,$ find the integers.

If the legs of an isosceles right triangle are each 12 inches, how long, to the nearest one-thousandth of an inch, is the hypothenuse?

The hypothenuse and one leg of a right triangle are 18 and 12 inches respectively, how long, to the nearest one-thousandth of an inch is the other leg?

A ball is thrown vertically upward from the ledge of a building 75 feet above ground. The ball's height $h$ in feet above the ground at time $t$ in seconds is given by the equation$$h=-16 t^{2}+80 t+75.$$(a) how long does it take the ball to reach a height of 90 feet? (b) How long before the ball is back to its original position (at 75 feet)? (c) How long before the ball hits the ground? Give each answer to the nearest one-thousandth of a second.

An object is dropped from a helicopter hovering at 250 feet above the ground. The objects height in feet, $h$ is given by the equation $h=-16 t^{2}+250,$ where $t$ is measured in seconds. How long before the object is (a) 100 feet above the ground? (b) 50 feet above the ground? (c) on the ground? Give each answer to the nearest one-thousandth of a second.

A rectangular swimming pool is 30 feet by 40 feet. If a rectangular strip of grass of uniform width is to go around the pool, and the area of this strip is 624 square feet, how wide is the strip?

Two cars leave an intersection at the same time, one goes north and the other goes east. Some time later they are 125 miles apart. If the car moving north traveled 12 miles more than the one going east, how many far (to the nearest mile) did each car travel?

Barbara wants to purchase an area rug for her dining room whose dimensions are 20 feet by 24 feet. If the rectangular rug she purchases has an area of 216 square feet and is placed an equal distance from each wall (a) how wide is the uniform strip of uncovered flooring? (b) what are the dimensions of the rug?

John bikes a distance of 120 miles and then returns over the same route. On his return his average speed is 2 miles per hour more than when going. If the combined time for both trips was 22 hours, what was his speed each way?

Mary can build a computer in two hours less time than Tim. Working together, they can build a computer in 2 hours and 24 minutes. How long does it take Mary alone to build a computer?

The profit in dollars in producing $x$ -items of some commodity is given by the equation $P=-2 x^{2}+400 x-15000 .$ (a) How many items should be produced to break even? (b) How many items should be produced to maximize profit? (c) What is the maximum profit?

The profit in dollars in producing $x$ -items of some commodity is given by the equation $P=-20 x^{2}+1300 x-15000 .$ To the nearest integer, how many items should be produced to (a) yield a profit of $\$ 2,000 ?$ (b) break even?

Solve each of the following equations for the real values of $x$.$$\frac{12}{x-4}+\frac{18}{x+4}=\frac{9}{2}$$

Solve each of the following equations for the real values of $x$.$$\frac{18}{2 x+3}-\frac{8}{x+5}=1$$

Solve each of the following equations for the real values of $x$.$$\frac{4 x}{3 x-2}+\frac{16-3 x}{2 x+2}=2$$

Solve each of the following equations for the real values of $x$.$$x^{4}+5 x^{2}-36=0$$

Solve each of the following equations for the real values of $x$.$$6 x^{6}-17 x^{3}+12=0$$

Solve each of the following equations for the real values of $x$.$$x^{\frac{2}{3}}-35 x^{\frac{1}{3}}+216=0$$

Solve each of the following equations for the real values of $x$.$$2 x^{4}-4 x^{2}-3=0$$

Here is another proof of the quadratic formula. Begin with $a x^{2}+b x+c=0,$ and multiply each term of the equation by 4a. This gives $4 a^{2} x^{2}+4 a b x+4 a c=0 .$ Rewrite the equation as $4 a^{2} x^{2}+4 a b x=-4 a c$and add $b^{2}$ to each side giving $4 a^{2} x^{2}+4 a b x+b^{2}=b^{2}-4 a c$. Now factor the left-hand-side of this equation and complete the proof.

Given the quadratic equation $a x^{2}+b x+c=0$ and $c x^{2}+b x+a=0$ prove that the roots of one equation are the reciprocals of the roots of the other equation.